Turbulent Convection Anisotropies of Cosmic Rays

Yiran Zhang, Siming Liu

School of Physical Science and Technology, Southwest Jiaotong University

PASW@HUST, Wuhan, Mar 29, 2025

Outline

• Multiscale anisotropies of cosmic rays (CRs)
• Standard diffusion model
• Nonuniform convection anisotropy
• Regular nonuniform convection
• Turbulent convection
• Effect of background magnetic field
• Summary

Multiscale Anisotropies of CRs

• Highly isotropic angular distribution of relativistic charged particles
• Relative anisotropy (TeV) ~ 0.1%, dominated by a dipole moment ∝ E^δ, with δ ~ 1/3
• Highly relaxed particle trajectories: diffusive propagation

HAWC & IceCube 2019 ApJ

ℓ > 0

(10 TeV)

Multiscale Anisotropies of CRs

• Small-scale anisotropy problem
• Besides large-scale (dipole & quadrupole) structures, CRs have small-scale anisotropies, which cannot be explained by the standard diffusion model
• Angular power spectrum C_ℓ (TeV) ∝ ℓ^−α, with α ~ 2.7

HAWC & IceCube 2019 ApJ

(10 TeV)

ℓ > 3

(10 TeV)

Multiscale Anisotropies of CRs

• Modeling multiscale anisotropies
• Classical electromagnetism
• Nondiffusive transport (insufficient scattering)
• Simulation of particle motion in concrete magnetic fields (Schwadron et al. 2014 Science)
• Magnetic turbulence within mean free paths (power-law C_ℓ; Ahlers 2014 PRL)
• Diffusion approximation (sufficient scattering)
• Dipole: uniform pitch-angle diffusion (Ahlers 2016 PRL), Compton-Getting (CG; Compton & Getting 1935 PR) effect, fluid inertia (Zhang et al. 2022 ApJ)
• Quadrupole: nonuniform pitch-angle diffusion (Giacinti & Kirk 2017 ApJ), fluid shear (Zhang et al. 2022 ApJ)
• Small-scale structures: nonuniform pitch-angle diffusion (Malkov et al. 2010 ApJ), turbulent convection (power-law C_ℓ; Zhang & Liu 2024 ApJL)
• Beyond the standard model
• Neutral strangelets (Kotera et al. 2013 PLB)
• Dark matter annihilation (Harding 2013 arXiv)

Standard Diffusion Model

• Scattering due to magnetic field irregularities
• Resonance between charged particles (CRs) and plasma (hydromagnetic) waves with wave lengths comparable to the gyroradii r_g
• Pitch-angle scattering along background magnetic field lines gives rise to anisotropic spatial diffusion, with a scattering time τ ∝ p^2−γg (quasilinear theory)
• Bohm regime: isotropic diffusion (ID) when magnetic irregularities is comparable to the background field
• Dipole anisotropies
• Nonuniformity of sources yields a diffuse dipole anisotropy ∝ diffusion flux
• CG effect: uniform relative motion between the observer and scattering centers leads to a dipole anisotropy independent of CR energy
• No anisotropy smaller than the dipole scale can be produced by the standard diffusion model (uniform pitch-angle diffusion + uniform convection)

Nonuniform Convection Anisotropy

• Diffusion model with nonuniform convection
• Relaxation: after scattering, particles in the rest frame of the scattering center tend to completely lose information about their initial states, and have an isotropic distribution f
• Convection: co-movement of particles with the scattering center
• Relative motion of scattering centers at different locations leads to a nonuniform convection velocity U
• Backtracking particle trajectories
• Momentum change over the free path Δr (reference frame arguments)

Δp = −pΔU/v

• Distributional fluctuation or anisotropy (Liouville’s theorem; BGK analysis)

Δf = −Δr ∙ ∇f − Δp ∙ ∂f/∂p

Regular Nonuniform Convection

• Assuming ID for simplicity
• The free path is aligned with the particle arrival direction (Δr = τv = λ)
• Convection follows the background flow (plasma wave; U = u)
• Regular flow
• “Regular”: the free path is small (λ ≪ the flow scale Λ)
• Inertial and shear effects

Δu = (τu + λ) ∙ ∇u ~ (u/v + 1)uλ/Λ

• Convection anisotropies in a regular CR flow (∂lnf/∂lnp ≈ −4.7)
• CG dipole anisotropy ~ u/v, independent of energy
• Inertial dipole anisotropy ~ (u/v)²λ/Λ ∝ p^2−γg
• Shear quadrupole anisotropy ~ (u/v)λ/Λ ∝ p^2−γg, having the potential to explain the observation

Regular Nonuniform Convection

• CR shear anisotropy
• The observed quadrupole amplitude (> PeV) requires a shear rate ~ 0.03 Myr⁻¹ ~ Oort constants, for a diffusion coefficient (PeV) ~ 10³¹ cm²/s
• Galactic CR differential rotation? (consistent with the absence of the CG effect)

Zhang et al. 2022 ApJ

Turbulent Convection

• Generalization of the nonuniform convection scenario
• The Taylor expansion of the flow implies a high-order cutoff factor (λ/Λ)^l, which controls the existence of small-scale anisotropies
• The fluid description is valid only if relaxation is established (Λ ≳ λ)
• A regular flow has only the leading-order term (Λ ≫ λ), producing no anisotropy smaller than the quadrupole scale
• The cutoff factor can be neutralized if the flow property varies irregularly between any two spatial points (Λ ~ λ), implying that significant small-scale anisotropies may survive in a turbulent flow
• There is no fixed shape of turbulent convection anisotropies; we can only predict the ensemble behavior over realizations

Turbulent Convection

• Multipole anisotropies
• CR anisotropies are analyzed usually with the multipole (spherical harmonic) expansion, not the Taylor one
• The two-mode correlation of multipoles of the flow is determined by an integral transformation of the turbulence spectrum w (k) ∝ k^−γ, with the kernel being the bilinear of spherical Bessel functions
• The nonuniform convection anisotropy (p/p) ∙ (ΔU/v)∂lnf/∂lnp is a coupling of dipole and multipole moments
• Again, we can assume ID (λ/λ = p/p, U = u) for convenience
• For simplicity, we also assume an equipartition turbulent flow (a diagonalized two-point correlation tensor of Fourier components of ΔU), which gives rise to statistically isotropic distributional fluctuations (i.e., the two-mode correlation matrix of multipoles of Δf is diagonalized, with the elements being C_ℓ)

Turbulent Convection

• Geometric interpretation
• The angular scale of the 2^ℓ-pole structure is 2π/ℓ, corresponding to a length scale 2πλ/ℓ, with a wavenumber difference 1/λ from the 2^ℓ+1-pole
• The ensemble averaged angular power spectrum in a turbulent CR flow is

C_ℓ ~ w (ℓ/λ)/(v²λℓ) ∝ p^{(γ−1)(2−γg)}ℓ^−γ−1

• Exact solution

Turbulent Convection

• Convection & diffusion of TeV CRs
• Diffusion coefficient (TeV) ~ 10²⁹ cm²/s
• Kolmogorov law (γ = 5/3)
• (Alfvén) Velocity dispersion (10 pc) ~ 20 km/s

Zhang & Liu 2024 ApJL

Han 2017 ARAA

Lee & Lee 2019 NA

πλ (10 TeV)

πλ (10 TeV)/15

Effect of Background Magnetic Field

• Validity of ID
• The TeV diffusion coefficient of 10²⁹ cm²/s is from studies of the CR energy spectrum and diffuse dipole anisotropy (Zhang et al. 2022 MNRAS)
• The rule-of-thumb diffusion coefficient corresponds to λ (TeV) ~ 3 pc ≫ r_g (TeV) ~ 60 AU, implying that spatial diffusion of TeV CR should in fact be anisotropic
• The observed alignment of the TeV dipole anisotropy and local interstellar magnetic field (Schwadron et al. 2014 Science) also implies a nondiagonal diffusion tensor
• In principle, the diffusion tensor can be diagonalized via an ensemble of background magnetic fields, but such an ensemble is unlikely to be involved in existing observations

Effect of Background Magnetic Field

• Anisotropic spatial diffusion
• The free path is no longer aligned with the particle momentum (λ/λ ≠ p/p)
• Not all modes of magnetic irregularities are effective scattering centers, so the convection velocity is not identical to the background flow velocity (U ≠ u)
• Parallel diffusion (PD) model
• Ignoring the perpendicular transport, given that magnetic irregularities on TeV gyroresonance scales (2πr_g) are small
• The free path is purely along background magnetic field lines (Δr = λ ∙ BB/B²)
• Convection is also along the field lines (U = u ∙ BB/B²)
• The system is cylindrically symmetric
• Assuming uniform pitch-angle diffusion (p/p-independent λ) for simplicity

Effect of Background Magnetic Field

• PD & ID models should have a similar slope of angular power spectra in log-log space

Effect of Background Magnetic Field

• Difference between PD and ID models
• For given isotropic turbulence, PD gives a smaller overall amplitude of anisotropies (velocity dispersion (10 pc) ~ 50 km/s for fitting the observation)
• Distributional fluctuations in PD are not statistically isotropic

Preliminary

Summary

• The nonuniform convection scenario provides a simple framework within the classical spatial diffusion approximation to model the CR anisotropy on any angular scale
• The shear effect due to Galactic differential rotation may contribute to the CR quadrupole anisotropy at PeV energies
• The CR angular power spectrum may be an imprint of the interstellar Kolmogorov turbulence spectrum on the microscopic distribution
• Angular power spectra in ID and PD models have the same spectral index in the small-scale limit, though their distributional fluctuations have different statistical symmetry properties; the anisotropic one may carry information about the local background magnetic field